#8889329

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#36723	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(3,7)(5,8)(9,10,14,13)(11,12,16,15)(17,18,23,26)(19,24,21,20)(22,30,31,25)(27,28,29,32);(1,2,4,6)(3,5,7,8)(10,13)(11,16)(17,18,23,26)(19,24,21,20)(22,30,31,25)(27,28,29,32);(1,3)(2,5)(4,7)(6,8)(9,11,14,16)(10,15,13,12)(17,19,23,21)(18,20,26,24)(22,27,31,29)(25,28,30,32);(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,30)(31,32);(1,2)(3,5)(4,6)(7,8)(9,13,14,10)(11,12,16,15)(17,21,23,19)(18,24,26,20)(22,29,31,27)(25,32,30,28);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,22,23,31)(18,25,26,30)(19,27,21,29)(20,28,24,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796525);